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Here is the problem: derivative of (arctan(x squared plus x plus 1))(ln(e to the power x plus 1)).

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Find the derivative of (atan(x squared plus x plus 1)) (ln(e to the power x plus 1)) with respect to x.

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Product rule: (uv) prime equals u prime v plus uv prime.

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Chain rule — d by d x[ln u] equals u prime /u.

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Sum rule: differentiate each term separately.

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Exponential rule: d by d x[eᵘ] equals eᵘ times u prime.

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Chain rule — d by d x[arctan u] equals u prime /(1 plus u squared).

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Sum rule: differentiate each term separately.

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Sum rule: differentiate each term separately.

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Power rule: d by d x[x to the power n] equals n times x to the power n minus 1.

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Combine and simplify.

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So the answer is f prime of x equals (2x plus 1) times ln(e to the power x plus 1) / ((x squared plus x plus 1) squared plus 1) plus atan(x squared plus x plus 1) times e to the power x / (e to the power x plus 1).

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The derivative is built from the outside in: sums split into terms, constants factor out, products and quotients use their rules, and every composite function multiplies by the derivative of its inside (chain rule).

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Cross-checked numerically against an independent symbolic differentiator at 8 sample points, which checks out.